## Abstract

Spinning discs with liquid film on the surface have numerous applications in many industrial processes and have attracted a lot of investigations. Centrifugal atomization of metallic melts using a spinning disc is an efficient process for powder production and spray deposition, which is a typical example of a spinning disc interacting with a liquid film. In this paper, the vibration of an atomizing disc excited by a moving melt is analysed and the role of vibration in the disintegration of the melt film on atomizing discs is then investigated. A dynamic model of the atomizing disc as a spinning Kirchhoff plate is established with moving melt film treated as a moving load on the disc and as an unstable growing wave interacting with the surrounding air outside the disc. The powder size is analysed theoretically and then compared with experimental results. It is found that the predicted results agree with the experimental results very well in the film disintegration mode. Furthermore, the influences of the atomizer parameters on the melt break up and powder size are discussed. The control parameters in the centrifugal atomization are identified, which can provide guidance for atomizer designs.

## 1. Introduction

A spinning disc interacting with a thin liquid film on the surface has attracted numerous investigations in science and engineering because of its various applications in many industrial processes that range from chemical reactors to powder production in metallurgy (Uma & Usha 2009). Atomizing disc is an important device in metallurgy, using a spinning disc for powder production and spray deposition. It is a typical example of spinning discs interacting with liquid on the surface including complex physics processes and multi-disciplines. In the manufacturing process, a metallic melt flows from a nozzle onto the centre of the atomizing disc which is spinning at high speed. It then spreads quickly to the edge of the disc because of the large centrifugal force and induces moving weight, friction, thermal stress and pressure to the disc. At the edge of the disc, the melt is thrown off the disc and disintegrated into a spray of droplets with vast heat transfer in subsequence, which solidifies at room temperature to form powders with a narrow size distribution. The investigation of the atomizing disc is fundamental, intriguing and challenging, and may provide some insights into the general mechanical behaviour of rotating discs interacting with the flow of a thin fluid film on the surface. Furthermore, the study of the atomizing disc has great value in engineering. The technology of centrifugal atomization employing an atomizing disc enables the production of narrow powder size distributions with low energy cost. This technology was originally developed by the Pratt & Whitney Aircraft Group to produce Ni-based superalloy powders (Lawley 1992) and is widely used for manufacturing high-quality powders and near-net-shape preforms of a range of advanced metallic materials (Angers *et al.* 1997; Ozturk & Arslan 2001).

A number of investigations of flow behaviour of a liquid film on the surface of a spinning disc have been published since the pioneering work of Emslie *et al.* (1958) on fluid distribution of a Newtonian liquid on a rotating disc (Peurrung & Graves 1991; Myers & Charpin 2001). In centrifugal atomization, earlier studies focused on the flow behaviour on spinning discs, powder morphology and powder sizes. Previous research has shown that there are three modes of flow behaviour in the processes of centrifugal atomization (Fraser *et al.* 1963). At a low volumetric flow rate, direct droplet formation is observed at the edge of the disc. With increasing volumetric flow rates, there is a transition from direct droplet formation to ligament disintegration. Film disintegration emerges at even higher flow rates and the surface of the atomizing disc is covered by a continuous film moving from the centre to the circumference of the disc. In the direct droplet formation regime, the powder sizes as a function of processing parameters and melt properties have been calculated (Singer & Kisakurek 1976).

When ligament and film disintegration modes become the predominant mechanisms, however, theoretical prediction of powder size distribution seems impossible owing to the complexity of the physical processes. In practice, ligament and film disintegration modes are the more popular working modes in centrifugal atomization because of their high efficiency of powder production. The break-up mechanisms of the film disintegration mode embrace the break-up mechanisms of other disintegration modes, which is recognized by many researchers. Thus, the investigation of the film disintegration mode is the basis of other disintegration modes and provides understanding of the physical processes involved in centrifugal atomization. Usually, empirical correlations for some specific melts in a certain range of atomizing conditions were employed to express the powder size distribution (Labrecque *et al.* 1997). The derived correlations are applicable for fairly narrow variations of geometry and process parameters. Many of the empirical correlations for metal powder sizes fail to provide quantitative information on mechanisms of powder formation. Zhao and his co-workers (Zhao *et al.* 1998; Zhao 2004*a*,*b*) simulated the height distribution of the continuous axisymmetric film in centrifugal atomization based on Newtonian fluid mechanics. In the real case, the flow behaviour of the melt is usually very complicated, considering the interaction between the melt flow and the disc. Mathematical models can be used to calculate the thickness and velocity of the melt film and thus to predict or explain the experimental results, such as powder shape and size distribution (Li & Tsakiropoulos 2001; Liu & Li 2007). However, the models need to be experimentally validated before being used with confidence if the vibration of atomizing discs is not considered, because the vibration of the device definitely influences the flow behaviour and subsequently affects the quality of the powders.

Although vibration of spinning discs such as compact discs (CDs), circular saws and brakes has been extensively investigated (Ouyang & Mottershead 2001; Tian & Hutton 2001; Kirpekar & Bogy 2008), the investigations of the vibration of atomizing discs are rarely reported in the open literature. The vibration of atomizing discs was first studied by Ouyang (2005, 2007). The atomizing disc was modelled as a spinning Kirchhoff plate and the continuous film as a moving distributed mass in the radial direction. Besides the simplistic dynamic model of atomizing discs, the role of vibration of the atomizing disc in centrifugal atomization is still uncertain. It is recognized that the higher the disc rotation, the finer the powers but greater vibration and louder noise will result. The disc vibration is undesirable owing to its contribution to potential fatigue failure of the device and generation of loud noise that may be detrimental physically and psychologically to nearby workers. Disc vibration and noise imposes a limit to the size of powders produced by centrifugal atomization. On the other hand, melt disintegration and powder formation in centrifugal atomization may benefit from the disc vibration, which has not been identified.

In this paper, the vibration of an atomizing disc is investigated and a theoretical model using the vibration results to predict the powder size produced by the atomizing disc is presented. A dynamic model of the atomizing disc as a spinning Kirchhoff plate subjected to moving melt film is established. The hydraulic jump encountered in the moving film prior to the disintegration, which was ignored by Ouyang’s work, is considered in this dynamic model. The melt film outside the disc is then treated as a growing wave under the initial disturbance caused by the disc vibration. A theoretical model for stability analysis of this kind of melt film, which is modelled as a viscous, incompressive moving sheet wave in inviscid, incompressive surrounding air is presented and solved numerically. Finally, based on the results of disc vibration, several examples are analysed for the powder size prediction. It is found that the predicted results of powder size agree with the experimental results very well in the film disintegration mode, which indicates that the presented model is applicable in the film disintegration mode. Furthermore, the correlations between the size of powder generated by atomizing discs and a variety of parameters, which can provide a guidance for atomizer designs, are presented in this paper. The control parameters in the centrifugal atomization are identified and some of those important ones, which have not been recognized in previous work, are clarified in this paper.

## 2. Dynamic model of atomizing discs

In centrifugal atomization, as shown in figure 1, the metallic melt descends to the centre of the atomizing disc spinning at a high speed and is accelerated towards the edge of the disc prior to disintegration. The load on the atomizing disc is complicated, involving the gravity of the moving film, fluid inertial force, friction, thermal stress, and so on. To establish the dynamic model, several assumptions are made in this paper: (i) the distribution of the melt film is axisymmetric and does not vary in the circumferential direction; (ii) the friction between the film and the disc can be ignored; (iii) the thermal deformation is negligible; (iv) the influence of the vibration of the disc on the film height is assumed to be negligible; (v) the interaction between the disc and the surrounding air is assumed to be negligible. This last assumption may not be very valid. However, this assumption has to be made to avoid extra complexity at this stage of the investigation and it will be removed in the next stage.

The atomizing disc can be modelled as a spinning Kirchhoff plate, while the melt film can be treated as a moving load travelling in the radial direction. The equation of motion of the spinning disc subjected to the external distributed load *p* is2.1where the biharmonic differential operator written in the cylindrical coordinate system is2.2where *ρ*, *h*, *D*, *ν*, *a*, *b* are the mass density, the thickness, the flexural rigidity of the plate, Poisson’s ratio, inner radius and outer radius. It must be noticed that this equation is established in the rotating frame fixed to the spinning disc at speed *Ω*. Dimensionless variables are defined by2.3

Under definition (2.3), equation (2.1) becomes2.4

The in-plane stresses *σ*_{θd} and *σ*_{rd} in equation (2.4) (Iwan & Moeller 1976) are:2.5where2.6where *a*_{c}=*a*/*b* is the clamping ratio of the atomizing disc.

In the region where the melt is present on the disc surface, the external distributed load has two sources: one is due to the fluid inertia force and the other due to the weight of the film.2.7where2.8

Under the definition of equation (2.3), equation (2.7) becomes2.9where *ρ*_{md}=*ρ*_{m}/*ρ*, *H*_{m}=*h*_{m}/*h* and *R*_{m}=*r*_{m}/*b*.

## 3. Height distribution of the melt film prior to disintegration

The melt flow under centrifugal force on the surface of the atomizing disc can be assumed as axisymmetric-continuous in the film disintegration mode. Assuming the melt as Newtonian fluid and neglecting the effects of heat transfer between the melt and the disc, the motion of melt film can be determined by the Navier–Stokes equations of momentum conservation and mass conservation as follows:3.1
3.2
3.3and
3.4where *U*_{m}(*r*), *U*_{m}(*θ*), *U*_{m}(*z*) are the velocities of the moving melt film in the radial direction, tangential direction and vertical direction, respectively; *p*_{m} is the pressure, *ρ*_{m} is the density, *ν*_{m} is the kinematic viscosity of the melt, is a mathematical operator which is defined as3.5

The radial velocity and the height distribution of the melt from the above equations with appropriate boundary conditions are shown as follows: (Zhao *et al.* 1998)3.6and3.7where *ν*_{m} and *Q* are kinematic viscosity and volume flow rate of the melt, constants *n*_{c}, *n*_{k} and *n*_{b} are 0.702, 0.587 and 0.739, respectively, *U*_{w0} and *r*_{0} are the initial velocity of the melt and the initial radius of the melt film, *r*_{c} is critical jump radius where hydraulic jump happens. Hydraulic jump is a phenomenon where the height of the melt dramatically rises at a radius prior to the disintegration (Zhao 2004*a*,*b*). The melt film keeps flowing and becomes unstable after being thrown off the atomizing disc.

## 4. Solutions

First of all, the modes of the disc under the influence of the centrifugal force must be determined. It is well known that the modes of a stationary disc are a combination of Bessel functions. However, when the stresses caused by the centrifugal force are present, there is no closed-form solution to the resultant eigenvalue equation from equation (2.4). Here, a Galerkin’s method is used to obtain the modes of the rotating disc, which are also a combination of Bessel functions, in the form of **a**^{T}** Ψ**(

*R*), where4.1In equation (4.1),

*ψ*

_{i}(

*R*)=

*R*

_{mn}(

*R*) is the combination of the Bessel functions that satisfy the boundary conditions. As this is a fairly standard procedure, it is not shown here. Now, the solution of equation (2.1) can be written as4.2where i is . The modes of the disc are:4.3and4.4where

*J*

_{n}(

*kR*) and

*Y*

_{n}(

*kR*) are Bessel functions of the first and second kind of order

*n*, and

*I*

_{n}(

*kR*) and

*K*

_{n}(

*kR*) are modified Bessel functions of the first and second kind of order

*n*. The value of

*m*represents the number of nodal circles, while

*n*represents the number of nodal diameters in a disc vibration mode. The ratios of the constants

*α*,

*β*,

*γ*,

*δ*and the natural frequency parameter

*k*are determined by the boundary conditions of the disc (Lamb & Southwell 1921). The natural frequency parameter

*k*is given by 4.5in which

*ω*

_{mn}are the circular frequencies of the stationary disc.

By using a Galerkin method, equation (2.4) becomes4.6where4.7

Substituting equation (4.3) into equation (4.6) yields4.8

Because of orthogonality of modes4.9

One gets4.10

Substitution of equation (2.9) into equation (4.10) yields4.11Equation (4.11) represents a number of simultaneous ordinary differential equations in unknown modal coordinates *q*_{j}(*t*), to be solved by a Runge–Kutta method coded in Matlab. From *q*_{j}(*t*), the vibrating frequencies *f*_{j} of the disc subjected to the moving melt film can be extracted, to be shown in figure 6 in §6. With the definitions of dimensionless variables in equation (2.3), the relationship between the real frequency *f*_{j} and the dimensionless frequency *F*_{j} can be expressed by4.12

## 5. Instability of the melt film

The instability of liquid has been investigated for a number of years. A variety of mechanisms and theories such as Rayleigh mechanism (Strutt & Rayleigh 1878), Weber theory (Weber 1931) and Taylor instability theory (Taylor 1950) have been proposed. Extensive reviews about the instability of liquid have been made (Liu 1999; Malkin 2008). For the purpose of understanding melt film disintegration, the melt film can be considered as a moving sheet. The principal cause of the instability of the moving sheet is the interaction of the sheet with the surrounding atmosphere, whereby rapidly growing waves are imposed on the sheet (Dombrowski & Johns 1963). When the wave amplitude reaches a critical value, disintegration occurs and the sheet breaks up into unstable ligaments under the force of surface tension, pressure and centrifugal force. Subsequently, the unstable ligaments form droplets owing to the contracting action of the surface tension. In centrifugal atomization, if the observer sits on the spinning disc, the melt film is seen to flow radially when omitting the effect of Coriolis force and side slip between the film and the disc, which is found to be basically true (Myers & Charpin 2001). Under the disturbance of the vibration of the edge of the disc and the pull of the centrifugal force, the moving melt film forms a travelling wave. Figure 2 illustrates the side view of the growing wave of the melt film at the edge of the disc. It is important to note that the film behaves differently when on the disc and when moving in the air after leaving the disc, and hence is governed by different equations. The wave outside the disc discussed in the paper is mainly because of disturbance of the vibration of the disc and the interaction between the wave and the surrounding air. On the other hand, the vibration of the disc is assumed not to have much influence on the formation of wavy film surface on the disc. Sisoev *et al.* (2003) calculated the axisymmetric wave in viscous liquid film flowing over a spinning disc neglecting the vibration of the disc. The wave in the melt film on the disc, if it exists, can affect the film formation on the disc and cause the transition of disintegration modes, for instance, from film disintegration mode to ligament disintegration mode. In this paper, the discussion is focused on the film disintegration mode, so the possible presence of wave in the melt film on the disc is not considered. It would be exceedingly complicated and beyond the scope of the current paper if the influence of the disc vibration on the film formation on the disc is considered. This assumption will be re-examined in the next stage of this research.

This melt film travelling in the air at velocity *U*_{w}(*r*) is analogous to the sheet moving in the surrounding atmosphere. The melt and the air are assumed to be incompressible based on the fact that the velocity *U*_{w}(*r*) is much smaller than the velocity of sound. The wave motion of the melt film under the disturbance of vibration can be assumed as5.1where *A*_{m} and *k*_{m} are the wave amplitude and the wavenumber, and *A*_{m}=*A*_{0}e^{ωt}, in which *A*_{0} is the amplitude of initial disturbance of the melt film wave equalling the displacement amplitude of the vibration of the disc edge, and *ω* is a complex number represented by *ω*_{r}+i*ω*_{m}. The wavenumber *k*_{m} is determined by the wave velocity and the disturbance frequency, which is the frequency of the vibration *f*_{j}.

To investigate the instability of the melt film wave, the linearized liquid continuity and momentum equations subjected to the boundary conditions at the interfaces between the melt film and surrounding air can be expressed as follows:

Continuity requirement for melt:5.2

Momentum equilibrium for melt:5.3and5.4where *V*_{r} and *V*_{z} are the melt velocity components in the radial and vertical directions, ∇^{2} is the Laplacian operator. The boundary conditions at the two interfaces for the above equations are: (i) the kinematic-free surface condition; (ii) vanished shear stress at the surface; (iii) the continuity of normal stress. The mathematical statements of these boundary conditions can be expressed as:5.5
5.6
5.7and
5.8
where *τ*_{rz} and *σ*_{zz} are the shear and normal stresses of the melt film, *σ*_{gzz} is the normal stress of the air, *p*_{σ} is the pressure induced by surface tension.

To obtain the normal stress of air at the interfaces which induces the effect of the surrounding air on the instability of the melt film wave, the equations of motion for air should be solved. The air is assumed to be inviscid and stationary before the disturbance sets in.

Continuity requirement for air:5.9

Momentum equilibrium for air:5.10and5.11where *V*_{gr} and *V*_{gz} are the velocity components of air in the radial and vertical directions, *ρ*_{g} and *p*_{g} are the density and pressure of air.

The corresponding boundary conditions at the interfaces and far from the interfaces are:5.12and5.13

In order to solve the equations for melt film motion, the velocities of the melt are separated into two parts using the Helmholtz decomposition:5.14where *V*_{rI} and *V*_{zzI} are the irrotational solutions, while *V*_{rII} and *V*_{zII} are the solutions containing the effect of viscosity. These velocities can be expressed by a velocity potential *ϕ*_{m} and stream function *ψ*_{m}.

The pressure of the melt film is given by5.15

It is assumed that *ϕ*_{m} and *ψ*_{m} have the following forms with the melt film wave motion.5.16and5.17

Substitution of equations (5.16) and (5.17) into equation (5.14), with equation (5.15), the momentum equilibrium equations (5.3) and (5.4) can be simplified and reduced to5.18and5.19where

The solutions of equations (5.18) and (5.19) are5.20and5.21

From the boundary conditions represented by equations (5.5) and (5.6), we can solve the constants5.22and5.23

Using a similar approach and introducing the velocity potential *ϕ*_{g} to solve the equations of motion for surrounding air,5.24and5.25where5.26

Thus, the solution considering the boundary conditions of air is that5.27

From equations (5.8) and (5.15), the normal stress of melt film at the interfaces can be expressed as5.28

The normal stress of air at the interfaces is5.29

The pressure induced by surface tension is5.30

Substitution of equations (5.28), (5.29) and (5.30) into equation (5.7) yields the following dispersion relation between *ω* and *k*_{m}5.31

If viscosity is neglected, the above equation (5.31) reduces to5.32

This equation is very similar to the dispersion relation derived by Senecal *et al.* (1999) for a two-dimensional inviscid incompressive liquid sheet moving at high speed in an inviscid gas in sinuous mode. The differences are in the definition of *K*_{2} and the third block of terms in equation (5.32), because the coordinate system in their work is Cartesian coordinate system.

In order to predict the powder sizes, the physical mechanisms of sheet disintegration proposed by Dombrowski & Johns (1963), which have been recognized by many researchers are adopted. When *ω*_{r} is greater than zero, the amplitude of the melt film wave grows with time. If the amplitude reaches a critical value of *A*_{c}, the melt film breaks up into unstable fragments, which can be measured by the amplitude growth number defined by .

Once it is certain that the melt film would break up in the film disintegration mode, the following formulae can be used to predict the powder size. The diameter of the resulting ligaments from the unstable fragments of half wavelength obtained by a mass balance is given by5.33

The diameter of the droplet formed from the ligaments could be calculated by Weber theory (Weber 1931) as5.34where is known as the Ohnesorge number.

The diameter of the powders produced after solidification is calculated by a mass balance5.35where *ρ*_{s} is the density of the solid-state powders. From the above equations, it is obvious that the wavenumber of the melt film, which is dominated by the initial disturbance of the vibration of the disc edge, has a direct effect on the powder size. Thus, it is important to investigate the vibration of the atomizing disc.

## 6. Results and discussion

### (a) The amplitude growth of unstable melt film

The amplitude of the melt film wave keeps increasing if the amplitude growth ratio is greater than zero. When the amplitude growth number *β* reaches a critical value, 12, reported in previous investigations (Dombrowski & Hooper 1962), the melt film wave will be disintegrated into ligaments. The amplitude growth ratio and the amplitude growth number can be obtained based on the numerical calculation of equation (5.31). Figure 3 shows a typical numerical result of the amplitude growth number *β* and the amplitude growth ratio *ω*_{r}. The melt is tin while the surrounding gas is air. The properties of tin used are *ρ*_{m}: 6670 kg cm^{−3}, *μ*: 1.18×10^{−3} Pa⋅s, *σ*: 0.51 N m^{−1}. The rotating speed of the atomizing disc here is 6000 r.p.m. and the wavenumber is 10. As shown in figure 3, the amplitude increases with time. The amplitude growth ratio increases quickly until 0.1 s later.

As shown in figure 4, it is clearer if the horizontal time coordinate is changed to logarithmic coordinate. With time advancing, at first, the amplitude growth ratio *ω*_{r} dramatically increases; after a short time, between 10^{−4} and 10^{−2} s, the increasing speed of the amplitude growth ratio slows down at this stage. Around 0.1 s, the amplitude growth ratio becomes steady. The corresponding amplitude growth number, which is the integration of the amplitude growth ratio with time, keeps rising.

Equation (5.31) indicates that the instability of the melt film is related to two types of parameters. One type of parameter is the properties of the melt and the surrounding gas, such as density, viscosity and surface tension. The properties of the melt and the surrounding gas for a given atomizing system have a certain range of values which are temperature dependent. The other type of parameter is the thickness and travelling speed of the melt film. This type of parameter is the most influenced by the rotating speed of the atomizing disc and the flow rate of the melt. Figure 5 shows the amplitude growth ratio *ω*_{r} with various rotating speeds, which are 6000, 9000, 12 000 and 15 000 r.p.m. The melt flow rate is 220 kg h^{−1}. The arrow points to the direction of the increasing rotating speed. As illustrated in the figure, for a given rotating speed, the stable amplitude growth ratio increases with the wavenumber initially. After a certain point, the stable amplitude growth ratio then decreases with the wavenumber. This indicates that there is a maximum amplitude growth ratio at a wavenumber, which is defined as the optimum wavenumber in this paper. When the rotating speed increases, the unstable range of wavenumbers expands and the optimum wavenumber becomes larger. The powder size is inversely proportional to the wavenumber shown in equation (5.33) when the melt film breaks up into ligaments. This indicates that it is easier to get a smaller powder size with a higher rotating speed.

### (b) The frequency spectrum of atomizing discs

The above results indicate that the amplitude growth ratio curve acts like an amplifier. When the wavenumber of the melt film wave is close to the maximum amplitude growth ratio wavenumber, the melt film disintegrates most easily. The wavenumber is directly related to the frequency of the initial disturbance and the travelling speed. Figure 6 shows the frequency spectrum of the atomizing disc with variable rotating speeds. The properties of the atomizing disc are as follows. Young’s modulus is 210 GPa; Poisson’s ratio is 0.28; density of the disc is 7860 kg m^{−3}; the diameter of the atomizing disc is 80 mm; the thickness is 5 mm; the rotating speeds of the atomizing disc here are 6000, 9000, 12 000 and 15 000 r.p.m. As shown in figure 6, there is one dominant frequency with many minor peaks in the spectrum. The arrow shows the direction of the increase of the rotating speeds. When the rotating speed is increasing, the dominant frequency moves towards right in the spectrum. The reasons for this phenomenon are the following. The rotating speed induces the centrifugal force, which stiffens the atomizing disc in the rotating frame. Moreover, as shown in equation (3.7), the height of the melt film decreases with high rotating speed. Higher rotating speed means less melt mass on the disc, which also makes the frequencies of the atomizing disc increase.

### (c) Powder sizes prediction

Figure 7 presents the comparison between the predicted theoretical results of powder size and the mean experimental values reported in the work of Xie *et al.* (2004). The experiment used pure tin as the model metal at a temperature of 550^{°}*C*. The atomizing disc used in the experiment is described in the above section. When the melt flow rate is 220 kg h^{−1}, as shown in the figure, the predicted powder size agrees with the experimental data very well. Powder size decreases with the increase of the rotating speed. This is because the size of the powders of a certain melt in film disintegration mode, as shown in equations (5.34) and (5.35), is directly related to the wavenumber and the height distribution of the melt film. When the rotating speed of the atomizing disc rises, the optimum wavenumber, as discussed before, increases while the the height of the melt film decreases, which makes the size of the powder smaller as shown in the equations. For the same reason, the reduction of the melt flow rate also leads to finer powders.

It must be pointed out that the model presented in this paper is only suitable for film disintegration mode. There is a transition between film disintegration mode and ligament disintegration mode at higher rotating speeds or lower melt flow rates. Equation (6.1) is an empirical equation to predict the critical melt flow rate at the transitions among three modes (Champagne & Angers 1984).6.1where *K*_{c} is a constant, which is 0.07 at the transition from direct droplet mode to ligament disintegration mode, and 1.33 at the transition from ligament disintegration mode to film disintegration mode, *D*_{d} is the diameter of the atomizing disc. Figure 8 illustrates that when the melt flow rate is 150 kg h^{−1}, which corresponds to the ligament disintegration mode, there is a big gap between the theoretical results and experimental data in this situation indicating that the proposed model for powder prediction in this paper is not suitable for ligament disintegration mode in centrifugal atomization.

In practice, the information for the atomizer designs in centrifugal atomization, for metal powder production in particular, is lacking in the public domain because of the absence of adequate theoretical and empirical knowledge. The geometry and operating conditions of the atomizer are often chosen on the basis of the ease of fabrication and experience. Zhao (2006) provided a guidance for the design of a centrifugal atomizer for metal powder production based on energy analysis and previous work. He pointed out that the most important parameters to be considered in the atomizer designs are diameter, rotating speed and geometry. However, the guidance for working conditions and the correlations between the selected parameters and powder size are still not clear enough in the absence of understanding of the physical processes involved.

In the present work, the correlations between powder size and a variety of parameters are summarized in figure 9, which can provide useful information for centrifugal atomizer designs. From equations (5.33) to (5.35), the powder size can be expressed as6.2where is given in equation (4.12).

The above equation demonstrates that the powder size of the film disintegration mode is determined by the wavenumber, the thickness and the material properties of melt film. The wavenumber of the melt film is determined by the frequencies of the vibration of the atomizing disc and the travelling speed of the melt film wave. To get finer powders for a specific melt, higher frequency and thinner melt film are two options according to equation (6.2). The rotating speed of the atomizing disc, the melt flow rate and the geometry of disc have a large influence on the travelling speed and the height of the melt film, which is shown in equations (3.6) and (3.7). The results illustrated in figure 6 show that a higher rotating speed normally means a higher frequency. That is why the rotating speed of the atomizing disc and the melt flow rate are the two parameters employed to control the powder size in industry. As indicated by equation (4.12), the disc material properties such as density, Young’s modulus and Poisson’s ratio also affect the frequencies of the atomizing disc and influence the powder size subsequently. This means the designer should not only consider the geometry of the disc in terms of thickness, inner radius and outer radius, but also take the disc material properties into account. For example, in order to get finer powders, it is possible to raise the Young’s modulus or reduce the density of disc and so on, as indicated by equation (4.12). The density of the surrounding gas can also influence the melt flow on the atomizing disc and the optimum wavenumber, and thus affect the size and morphology of powders, which should be pointed out. The details of such influences will be investigated in the near future.

## 7. Conclusions

The vibration of an atomizing disc is analysed numerically and its role in centrifugal atomization is investigated in this paper. A dynamic model of the atomizing disc as a spinning Kirchhoff plate subjected to moving melt film is presented. The model is solved by a Galerkin method. Under the disturbance of the vibration of the atomizing disc, the moving melt film is modelled as an unstable wave whose amplitude keeps on growing until disintegration. It is found that the optimum wavenumber that leads to the largest amplitude growth ratio increases with the rotating speed. It is shown that the dominant frequency of atomizing discs has a strong influence on the melt disintegration. The predicted powder size using vibration results agrees with the experimental results very well in film disintegration mode. It is also shown that the powder size decreases with increasing rotating speed and reducing melt flow rate. Furthermore, the influences of the atomizer parameters on the melt break-up and the powder size are discussed. Based on the discussion, the control parameters in the centrifugal atomization are identified. Some of those important ones which have not been recognized in previous work have been clarified in this work. The correlations between the size of powders generated by atomizing discs and those control parameters, which can provide guidance for atomizer designs, are proposed in this paper.

## Acknowledgements

The authors acknowledge the support of their colleagues in the Dynamics and Control Group at the University of Liverpool and the helpful discussions with Dr Y. Y. Zhao. Financial support to the first author by the China Scholarship Council (CSC), the ORSAS award from the Higher Education Funding Council for England and Duncan Norman Fellowship are gratefully acknowledged. The second author wishes to acknowledge the Royal Academy of Engineering/Leverhulme Trust Senior Fellowship.

- Received February 19, 2010.
- Accepted June 21, 2010.

- This journal is © 2010 The Royal Society